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Symmetry
Volume 15
Issue 8
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Symmetry, Volume 15, Issue 8 (August 2023) – 147 articles

Cover Story (view full-size image): The Standard Model (SM) of particle physics—augmented by weak-scale supersymmetry—is a highly motivated extension of the SM, which also predicts additional Higgs bosons that carry electric charge. Such new charged Higgs matter states can be produced at high rates at the CERN Large Hadron Collider, and they would then decay into both SM and new SUSY particle final states. Here, we present their production cross sections, decay rates and prospects for discovery or exclusion at the future high luminosity LHC experiments in the well-motivated case of natural SUSY models, which are thought to emerge from the string theory landscape of vacua. View this paper
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15 pages, 453 KiB  
Article
Graph Embedding with Similarity Metric Learning
by Tao Tao, Qianqian Wang, Yue Ruan, Xue Li and Xiujun Wang
Symmetry 2023, 15(8), 1618; https://doi.org/10.3390/sym15081618 - 21 Aug 2023
Viewed by 1766
Abstract
Graph embedding transforms high-dimensional graphs into a lower-dimensional vector space while preserving their structural information and properties. Context-sensitive graph embedding, in particular, performs well in tasks such as link prediction and ranking recommendations. However, existing context-sensitive graph embeddings have limitations: they require additional [...] Read more.
Graph embedding transforms high-dimensional graphs into a lower-dimensional vector space while preserving their structural information and properties. Context-sensitive graph embedding, in particular, performs well in tasks such as link prediction and ranking recommendations. However, existing context-sensitive graph embeddings have limitations: they require additional information, depend on community algorithms to capture multiple contexts, or fail to capture sufficient structural information. In this paper, we propose a novel Graph Embedding with Similarity Metric Learning (GESML). The core of GESML is to learn the optimal graph structure using an attention-based symmetric similarity metric function and establish association relationships between nodes through top-k pooling. Its primary advantage lies in not requiring additional features or multiple contexts, only using the symmetric similarity metric function and pooling operations to encode sufficient topological information for each node. Experimental results on three datasets involving link prediction and node-clustering tasks demonstrate that GESML significantly improves learning for all challenging tasks relative to a state-of-the-art (SOTA) baseline. Full article
(This article belongs to the Section Computer)
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<p>A graph with some edges and nodes.</p> Full article ">Figure 2
<p>GAP.</p> Full article ">Figure 3
<p>GESML.</p> Full article ">Figure 4
<p>Effect of the number of neighbors sampled on the Cora dataset.</p> Full article ">Figure 5
<p>Effect of the number of neighbors sampled on the Email dataset.</p> Full article ">Figure 6
<p>Effect of the number of neighbors sampled on the Zhihu dataset.</p> Full article ">Figure 7
<p>The effect of the number of sampled neighbors on the model’s performance.</p> Full article ">Figure 8
<p>The effect of various k on the model’s performance.</p> Full article ">
20 pages, 442 KiB  
Article
Novel Approach to Multi-Criteria Decision-Making Based on the n,mPR-Fuzzy Weighted Power Average Operator
by Tareq Hamadneh, Hariwan Z. Ibrahim, Mayada Abualhomos, Maha Mohammed Saeed, Gharib Gharib, Maha Al Soudi and Abdallah Al-Husban
Symmetry 2023, 15(8), 1617; https://doi.org/10.3390/sym15081617 - 21 Aug 2023
Viewed by 1363
Abstract
A significant addition to fuzzy set theory for expressing uncertain data is an n,m-th power root fuzzy set. Compared to the nth power root, Fermatean, Pythagorean, and intuitionistic fuzzy sets, n,m-th power root fuzzy sets can cover more uncertain situations due to [...] Read more.
A significant addition to fuzzy set theory for expressing uncertain data is an n,m-th power root fuzzy set. Compared to the nth power root, Fermatean, Pythagorean, and intuitionistic fuzzy sets, n,m-th power root fuzzy sets can cover more uncertain situations due to their greater range of displayed membership grades. When discussing the symmetry between two or more objects, the innovative concept of an n,m-th power root fuzzy set over dual universes is more flexible than the current notion of an intuitionistic fuzzy set, a Pythagorean fuzzy set, and a nth power root fuzzy set. In this study, we demonstrate a number of additional operations on n,m-th power root fuzzy sets along with a number of their special aspects. Additionally, to deal with choice information, we create a novel weighted aggregated operator called the n,m-th power root fuzzy weighted power average (FWPAmn) across n,m-th power root fuzzy sets and demonstrate some of its fundamental features. To rank n,m-th power root fuzzy sets, we also define the score and accuracy functions. Moreover, we use this operator to identify the countries with the best standards of living and show how we can select the best option by contrasting aggregate results using score values. Finally, we contrast the results of the FWPAmn operator with the square-root fuzzy weighted power average (SR-FWPA), the nth power root fuzzy weighted power average (nPR-FWPA), the Fermatean fuzzy weighted power average (FFWPA), and the n,m-rung orthopair fuzzy weighted power average (n,m-ROFWPA) operators. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—Volume II)
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<p>Several n,mPR-FS-type grade spaces.</p> Full article ">
13 pages, 1997 KiB  
Article
On the Solitary Waves and Nonlinear Oscillations to the Fractional Schrödinger–KdV Equation in the Framework of the Caputo Operator
by Saima Noor, Badriah M. Alotaibi, Rasool Shah, Sherif M. E. Ismaeel and Samir A. El-Tantawy
Symmetry 2023, 15(8), 1616; https://doi.org/10.3390/sym15081616 - 21 Aug 2023
Cited by 9 | Viewed by 1155
Abstract
The fractional Schrödinger–Korteweg-de Vries (S-KdV) equation is an important mathematical model that incorporates the nonlinear dynamics of the KdV equation with the quantum mechanical effects described by the Schrödinger equation. Motivated by the several applications of the mentioned evolution equation, in this investigation, [...] Read more.
The fractional Schrödinger–Korteweg-de Vries (S-KdV) equation is an important mathematical model that incorporates the nonlinear dynamics of the KdV equation with the quantum mechanical effects described by the Schrödinger equation. Motivated by the several applications of the mentioned evolution equation, in this investigation, the Laplace residual power series method (LRPSM) is employed to analyze the fractional S-KdV equation in the framework of the Caputo operator. By incorporating both the Caputo operator and fractional derivatives into the mentioned evolution equation, we can account for memory effects and non-local behavior. The LRPSM is a powerful analytical technique for the solution of fractional differential equations and therefore it is adapted in our current study. In this study, we prove that the combination of the residual power series expansion with the Laplace transform yields precise and efficient solutions. Moreover, we investigate the behavior and properties of the (un)symmetric solutions to the fractional S-KdV equation using extensive numerical simulations and by considering various fractional orders and initial fractional values. The results contribute to the greater comprehension of the interplay between quantum mechanics and nonlinear dynamics in fractional systems and shed light on wave phenomena and symmetry soliton solutions in such equations. In addition, the proposed method successfully solves fractional differential equations with the Caputo operator, providing a valuable computational instrument for the analysis of complex physical systems. Moreover, the obtained results can describe many of the mysteries associated with the mechanism of nonlinear wave propagation in plasma physics. Full article
(This article belongs to the Special Issue Nonlinear Differential and Integral Equations and Their Infinite Systems)
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<p>The profile of LRPSM periodic solution <math display="inline"><semantics> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>ϱ</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> in (<b>a</b>–<b>c</b>) three-dimensional and (<b>d</b>) two-dimensional form, at different values of fractional order.</p> Full article ">Figure 2
<p>The profile of LRPSM periodic solution <math display="inline"><semantics> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>ϱ</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> in (<b>a</b>–<b>c</b>) three-dimensional and (<b>d</b>) two-dimensional form, at different values of fractional order.</p> Full article ">Figure 3
<p>The profile of LRPSM soliton solution <math display="inline"><semantics> <mrow> <mi>w</mi> <mo stretchy="false">(</mo> <mi>ϱ</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> in (<b>a</b>–<b>c</b>) three-dimensional and (<b>d</b>) two-dimensional form, at different values of fractional order.</p> Full article ">
22 pages, 5596 KiB  
Article
A Free Convective Two-Phase Flow of Optically Thick Radiative Ternary Hybrid Nanofluid in an Inclined Symmetrical Channel through a Porous Medium
by K. M. Pavithra, Pudhari Srilatha, B. N. Hanumagowda, S. V. K. Varma, Amit Verma, Shalan Alkarni and Nehad Ali Shah
Symmetry 2023, 15(8), 1615; https://doi.org/10.3390/sym15081615 - 21 Aug 2023
Cited by 6 | Viewed by 1288
Abstract
In the present article, we investigate the free convective flow of a ternary hybrid nanofluid in a two-phase inclined channel saturated with a porous medium. The flow has been propelled using the pressure gradient, thermal radiation, and buoyancy force. The flow model’s governing [...] Read more.
In the present article, we investigate the free convective flow of a ternary hybrid nanofluid in a two-phase inclined channel saturated with a porous medium. The flow has been propelled using the pressure gradient, thermal radiation, and buoyancy force. The flow model’s governing equations are resolved using the regular perturbation approach. The governing equations are solved with the help of the regular perturbation method. Polyethylene glycol and water (at a ratio of 50%:50%) fill up Region I, while a ternary hybrid nanofluid based on zirconium dioxide, magnesium oxide, and carbon nanotubes occupies Region II. The ternary hybrid nanofluids are defined with a mixture model in which three different shapes of nanoparticles, namely spherical, platelet, and cylindrical, are incorporated. The consequences of the most significant variables have been examined using both visual and tabular data. The main finding of this work is that utilising a ternary hybrid nanofluid at the plate y = 1 increases the rate of heat transfers by 753%, demonstrating the potential thermal efficiency. The overall heat and volume flow rates are amplified by buoyant forces and viscous dissipations and dampened by the thermal radiation parameter. The optimum enhancement of temperature is achieved by the influence of buoyancy forces. A ternary nanofluid region experiences the maximum temperature increase compared to a clear fluid region. To ensure the study’s efficiency, we validated it with prior studies. Full article
(This article belongs to the Section Engineering and Materials)
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<p>Flow geometry.</p> Full article ">Figure 2
<p>The impact of <math display="inline"><semantics> <mrow> <mi>G</mi> <mi>r</mi> <mi>t</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>The impact of <math display="inline"><semantics> <mi>σ</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>The impact of <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>r</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>The impact of <math display="inline"><semantics> <mi>R</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>The impact of <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>1</mn> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>The impact of <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>The impact of <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mn>3</mn> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>The effect of <math display="inline"><semantics> <mrow> <mi>G</mi> <mi>r</mi> <mi>t</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>r</mi> </mrow> </semantics></math> on Nusselt number at <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>The effect of <math display="inline"><semantics> <mi>σ</mi> </semantics></math> and <math display="inline"><semantics> <mi>R</mi> </semantics></math> on Nusselt number at <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>The effect of <math display="inline"><semantics> <mrow> <mi>G</mi> <mi>r</mi> <mi>t</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>B</mi> <mi>r</mi> </mrow> </semantics></math> on skin friction coefficient at <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>The skin friction coefficient profiles against <math display="inline"><semantics> <mi>σ</mi> </semantics></math> and <math display="inline"><semantics> <mi>R</mi> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 13
<p>The impact of <math display="inline"><semantics> <mrow> <mi>G</mi> <mi>r</mi> <mi>t</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <mi>θ</mi> <mrow> <mo>(</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">
10 pages, 264 KiB  
Article
Symmetries of the Energy–Momentum Tensor for Static Plane Symmetric Spacetimes
by Fawad Khan, Wajid Ullah, Tahir Hussain and Wojciech Sumelka
Symmetry 2023, 15(8), 1614; https://doi.org/10.3390/sym15081614 - 21 Aug 2023
Viewed by 1012
Abstract
This article explores matter collineations (MCs) of static plane-symmetric spacetimes, considering the stress–energy tensor in its contravariant and mixed forms. We solve the MC equations in two cases: when the energy–momentum tensor is nondegenerate and degenerate. For the case of a degenerate energy–momentum [...] Read more.
This article explores matter collineations (MCs) of static plane-symmetric spacetimes, considering the stress–energy tensor in its contravariant and mixed forms. We solve the MC equations in two cases: when the energy–momentum tensor is nondegenerate and degenerate. For the case of a degenerate energy–momentum tensor, we employ a direct integration technique to solve the MC equations, which leads to an infinite-dimensional Lie algebra. On the other hand, when considering the nondegenerate energy–momentum tensor, the contravariant form results in a finite-dimensional Lie algebra with dimensions of either 4 or 10. However, in the case of the mixed form of the energy–momentum tensor, the dimension of the Lie algebra is infinite. Moreover, the obtained MCs are compared with those already found for covariant stress–energy. Full article
(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)
16 pages, 426 KiB  
Article
Implementing a Relativistic Motor over Atomic Scales
by Asher Yahalom
Symmetry 2023, 15(8), 1613; https://doi.org/10.3390/sym15081613 - 21 Aug 2023
Cited by 1 | Viewed by 1378
Abstract
A relativistic motor exchanging momentum and energy with an electromagnetic field is studied. We discuss the advantages and challenges of this novel mover, giving specific emphasis to the more favorable (yet challenging) nano configurations. It specifically turns out that an isolated hydrogen atom [...] Read more.
A relativistic motor exchanging momentum and energy with an electromagnetic field is studied. We discuss the advantages and challenges of this novel mover, giving specific emphasis to the more favorable (yet challenging) nano configurations. It specifically turns out that an isolated hydrogen atom in either a ground or excited state does not produce relativistic motor momentum. Full article
(This article belongs to the Special Issue Exploring Symmetry and Asymmetry in Astrophysics, Astronomy, and Space Science: Paving the Way for a Multiplanetary Future for Humanity)
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<p>Two current loops.</p> Full article ">Figure 2
<p>A relativistic engine.</p> Full article ">Figure 3
<p>A cross section of the relativistic engine.</p> Full article ">Figure 4
<p>The proximity between a classical electron and proton needed to achieve a desired velocity for an unloaded engine.</p> Full article ">Figure 5
<p>Two electrons from a train of electrons, moving in the vicinity of a proton.</p> Full article ">Figure 6
<p>Plasma of protons and electrons: the red circles symbolize protons, while the orange circles symbolize electrons.</p> Full article ">Figure 7
<p>The 100 planes of a lattice of <math display="inline"><semantics> <mrow> <mi>N</mi> <msup> <mi>a</mi> <mo>+</mo> </msup> <mi>C</mi> <msup> <mi>l</mi> <mo>−</mo> </msup> </mrow> </semantics></math> (table salt): blue circles depict sodium positive ions, and green circles depict chlorine negative ions. The trajectory of relativistic electrons is described using a thick black line.</p> Full article ">Figure 8
<p>A schematic of an elliptical orbit of an electron around a proton.</p> Full article ">
11 pages, 264 KiB  
Article
Saturated Varieties of Semigroups
by Muneer Nabi, Amal S. Alali and Sakeena Bano
Symmetry 2023, 15(8), 1612; https://doi.org/10.3390/sym15081612 - 21 Aug 2023
Viewed by 1290
Abstract
The complete characterization of saturated varieties of semigroups remains an unsolved problem. The primary objective of this paper is to make significant progress in this direction. We initially demonstrate that the variety of semigroups defined by the identity [...] Read more.
The complete characterization of saturated varieties of semigroups remains an unsolved problem. The primary objective of this paper is to make significant progress in this direction. We initially demonstrate that the variety of semigroups defined by the identity axy=ayxa is saturated. The next main result establishes that the variety of semigroups determined by the identity axy=ayax is saturated. Finally, we show that medial semigroups satisfying the identity xy=xyn, where n≥2, are also saturated. These results collectively lead to the conclusion that epis from these saturated varieties are onto. This paper thus offers substantial progress towards the comprehensive characterization of saturated varieties of semigroups. Full article
(This article belongs to the Special Issue Symmetry in Algebra and Its Applications)
11 pages, 5608 KiB  
Article
Symmetry and the Nanoscale: Advances in Analytical Modeling in the Perspective of Holistic Unification
by Paolo Di Sia
Symmetry 2023, 15(8), 1611; https://doi.org/10.3390/sym15081611 - 21 Aug 2023
Cited by 2 | Viewed by 1100
Abstract
Analytical modeling presents symmetries and aesthetic-mathematical characteristics which are not catchable in numerical computation for science and technology; nanoscience plays a significant role in unification attempts, considering also models including holistic aspects of reality. In this paper we present new discovered results about [...] Read more.
Analytical modeling presents symmetries and aesthetic-mathematical characteristics which are not catchable in numerical computation for science and technology; nanoscience plays a significant role in unification attempts, considering also models including holistic aspects of reality. In this paper we present new discovered results about the complete analytical quantum-relativistic form of the mean square deviation of position R2(t) related to a recently introduced Drude–Lorentz-like model (DS model), already performed at classical, quantum and relativistic level. The function R2(t) gives precise information about the distance crossed by carriers (electrons, ions, etc.) inside a nanostructure, considering both quantum effects and relativistic velocities. The model has a wide scale range of applicability; the nanoscale is considered in this paper, but it holds application from sub-pico-level to macro-level because of the existence of a gauge factor, making it applicable to every oscillating process in nature. Examples of application and suggestions supplement this paper, as well as interesting developments to be studied related to the model and to one of the basic elements of a current unified holistic approach based on vacuum energy. Full article
(This article belongs to the Special Issue Symmetry and the Nanoscale: Matter and Mind - Mathematical Modeling and Phenomenology)
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<p><math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> vs. <span class="html-italic">t</span> with <math display="inline"><semantics> <mrow> <msubsup> <mi>α</mi> <mrow> <msub> <mi>R</mi> <mrow> <mi>Q</mi> <mo>−</mo> <mi>R</mi> </mrow> </msub> </mrow> <mrow/> </msubsup> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>300</mn> <mi>K</mi> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <mi>v</mi> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mn>7</mn> </msup> <mrow> <mrow> <mi>c</mi> <mi>m</mi> </mrow> <mo>/</mo> <mi>s</mi> </mrow> </mrow> </semantics></math> (blue solid line), <math display="inline"><semantics> <mrow> <mi>v</mi> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>10</mn> </mrow> </msup> <mrow> <mrow> <mi>c</mi> <mi>m</mi> </mrow> <mo>/</mo> <mi>s</mi> </mrow> </mrow> </semantics></math> (red dashed line) and <math display="inline"><semantics> <mrow> <mi>v</mi> <mo>=</mo> <mn>2.5</mn> <mo>⋅</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>10</mn> </mrow> </msup> <mrow> <mrow> <mi>c</mi> <mi>m</mi> </mrow> <mo>/</mo> <mi>s</mi> </mrow> </mrow> </semantics></math> (green dot-dashed line).</p> Full article ">Figure 2
<p><math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> vs. <span class="html-italic">t</span> with <math display="inline"><semantics> <mrow> <msubsup> <mi>α</mi> <mrow> <msub> <mi>I</mi> <mrow> <mi>Q</mi> <mo>−</mo> <mi>R</mi> </mrow> </msub> </mrow> <mrow/> </msubsup> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>300</mn> <mi>K</mi> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <mi>v</mi> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mn>7</mn> </msup> <mrow> <mrow> <mi>c</mi> <mi>m</mi> </mrow> <mo>/</mo> <mi>s</mi> </mrow> </mrow> </semantics></math> (blue solid line), <math display="inline"><semantics> <mrow> <mi>v</mi> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>10</mn> </mrow> </msup> <mrow> <mrow> <mi>c</mi> <mi>m</mi> </mrow> <mo>/</mo> <mi>s</mi> </mrow> </mrow> </semantics></math> (red dashed line) and <math display="inline"><semantics> <mrow> <mi>v</mi> <mo>=</mo> <mn>2.5</mn> <mo>⋅</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>10</mn> </mrow> </msup> <mrow> <mrow> <mi>c</mi> <mi>m</mi> </mrow> <mo>/</mo> <mi>s</mi> </mrow> </mrow> </semantics></math> (green dot-dashed line).</p> Full article ">Figure 3
<p><math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> vs. <span class="html-italic">t</span> for state 1 with data from <a href="#symmetry-15-01611-t002" class="html-table">Table 2</a>.</p> Full article ">Figure 4
<p><math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> vs. <span class="html-italic">t</span> for state 2 with data from <a href="#symmetry-15-01611-t002" class="html-table">Table 2</a>.</p> Full article ">Figure 5
<p><math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> vs. <span class="html-italic">t</span> for state 3 with data from <a href="#symmetry-15-01611-t002" class="html-table">Table 2</a>.</p> Full article ">Figure 6
<p><math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> vs. <span class="html-italic">t</span> for the sum of the previous three states (data from <a href="#symmetry-15-01611-t002" class="html-table">Table 2</a>).</p> Full article ">Figure 7
<p><math display="inline"><semantics> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> vs. <span class="html-italic">t</span> for a state given by the sum of the previous three states (data from <a href="#symmetry-15-01611-t002" class="html-table">Table 2</a>), considering three different velocities (<math display="inline"><semantics> <mrow> <mi>v</mi> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mn>7</mn> </msup> <mrow> <mrow> <mi>c</mi> <mi>m</mi> </mrow> <mo>/</mo> <mi>s</mi> </mrow> </mrow> </semantics></math> (blue solid line), <math display="inline"><semantics> <mrow> <mi>v</mi> <mo>=</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>10</mn> </mrow> </msup> <mrow> <mrow> <mi>c</mi> <mi>m</mi> </mrow> <mo>/</mo> <mi>s</mi> </mrow> </mrow> </semantics></math> (red dashed line) and <math display="inline"><semantics> <mrow> <mi>v</mi> <mo>=</mo> <mn>2.5</mn> <mo>⋅</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mn>10</mn> </mrow> </msup> <mrow> <mrow> <mi>c</mi> <mi>m</mi> </mrow> <mo>/</mo> <mi>s</mi> </mrow> </mrow> </semantics></math> (green dot-dashed line)).</p> Full article ">
22 pages, 2119 KiB  
Article
On Performance of Marine Predators Algorithm in Training of Feed-Forward Neural Network for Identification of Nonlinear Systems
by Ceren Baştemur Kaya
Symmetry 2023, 15(8), 1610; https://doi.org/10.3390/sym15081610 - 20 Aug 2023
Cited by 3 | Viewed by 1105
Abstract
Artificial neural networks (ANNs) are used to solve many problems, such as modeling, identification, prediction, and classification. The success of ANN is directly related to the training process. Meta-heuristic algorithms are used extensively for ANN training. Within the scope of this study, a [...] Read more.
Artificial neural networks (ANNs) are used to solve many problems, such as modeling, identification, prediction, and classification. The success of ANN is directly related to the training process. Meta-heuristic algorithms are used extensively for ANN training. Within the scope of this study, a feed-forward artificial neural network (FFNN) is trained using the marine predators algorithm (MPA), one of the current meta-heuristic algorithms. Namely, this study is aimed to evaluate the performance of MPA in ANN training in detail. Identification/modeling of nonlinear systems is chosen as the problem. Six nonlinear systems are used in the applications. Some of them are static, and some are dynamic. Mean squared error (MSE) is utilized as the error metric. Effective training and testing results were obtained using MPA. The best mean error values obtained for six nonlinear systems are 2.3 × 10−4, 1.8 × 10−3, 1.0 × 10−4, 1.0 × 10−4, 1.2 × 10−5, and 2.5 × 10−4. The performance of MPA is compared with 16 meta-heuristic algorithms. The results have shown that the performance of MPA is better than other algorithms in ANN training for the identification of nonlinear systems. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Modelling: Topics and Advances)
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<p>Flowchart of MPA.</p> Full article ">Figure 2
<p>General structure of an artificial neuron.</p> Full article ">Figure 3
<p>Comparison of graphs belonging to real and predicted outputs for the training process.</p> Full article ">
11 pages, 1715 KiB  
Article
Stability Analysis of a Mathematical Model for Adolescent Idiopathic Scoliosis from the Perspective of Physical and Health Integration
by Yuhua Zhang and Haiyin Li
Symmetry 2023, 15(8), 1609; https://doi.org/10.3390/sym15081609 - 20 Aug 2023
Viewed by 1245
Abstract
In this paper, we take physical and health integration as the entry point. Firstly, based on the transformation mechanism of adolescent idiopathic scoliosis we construct a time delay differential model. Moreover, using the theory of characteristic equation we discuss the stability of a [...] Read more.
In this paper, we take physical and health integration as the entry point. Firstly, based on the transformation mechanism of adolescent idiopathic scoliosis we construct a time delay differential model. Moreover, using the theory of characteristic equation we discuss the stability of a positive equilibrium under the delays of τ=0 and τ≠0. Furthermore, through numerical simulation, it has been verified the delay, τ, exceeds a critical value, the positive equilibrium loses its stability and Hopf bifurcation occurs. Lastly, we determine that sports have a positive effect on adolescent idiopathic scoliosis, directly reducing the number of people with adolescent idiopathic scoliosis. Full article
(This article belongs to the Special Issue Mathematical Modeling in Biology and Life Sciences)
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<p>Transformation process between susceptible individuals <math display="inline"><semantics> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> and affected individuals <math display="inline"><semantics> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>The curves of system (15), where the blue and green curves represent <math display="inline"><semantics> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>, respectively.</p> Full article ">Figure 3
<p>The curves of <math display="inline"><semantics> <mi>t</mi> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mi>t</mi> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> in system (16) for <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, where the blue and green curves respresent <math display="inline"><semantics> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>, respectively.</p> Full article ">Figure 4
<p>The curves of <math display="inline"><semantics> <mi>t</mi> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mi>t</mi> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> in system (17) with <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>3.1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math>, where the blue and green curves respresent <math display="inline"><semantics> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>, respectively.</p> Full article ">Figure 5
<p>Orbits of <math display="inline"><semantics> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> for system (17) with <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>3.1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Orbits of <math display="inline"><semantics> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> for system (17) with <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>3.1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>2000</mn> </mrow> </semantics></math>.</p> Full article ">
18 pages, 875 KiB  
Article
A More Flexible Extension of the Fréchet Distribution Based on the Incomplete Gamma Function and Applications
by Jaime S. Castillo, Mario A. Rojas and Jimmy Reyes
Symmetry 2023, 15(8), 1608; https://doi.org/10.3390/sym15081608 - 20 Aug 2023
Cited by 3 | Viewed by 1505
Abstract
In this paper, a more flexible extension of the Fréchet distribution is introduced. The new distribution is defined by means of the stochastic representation as the quotient of two independent random variables, a Fréchet distribution and the power of a random variable, with [...] Read more.
In this paper, a more flexible extension of the Fréchet distribution is introduced. The new distribution is defined by means of the stochastic representation as the quotient of two independent random variables, a Fréchet distribution and the power of a random variable, with uniform distribution in the interval (0, 1). We will call this new extension the slash Fréchet distribution and one of its main characteristics is that its tails are heavier than the Fréchet distribution. The general density of this distribution and some basic properties are determined. Its moments, skewness coefficients, and kurtosis are calculated. In addition, the estimation of the model parameters is obtained by the method of moments and maximum likelihood. Finally, three applications with real data are performed by fitting the new model and comparing it with the Fréchet distribution. Full article
(This article belongs to the Special Issue Symmetry in Probability Theory and Statistics)
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<p>Graphical comparison of the density function of the Fréchet (Fr) and slash Fréchet (SFr) distributions for fixed alpha (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>) and different values of <span class="html-italic">q</span>.</p> Full article ">Figure 2
<p>Graphical comparison of the CDF between the Fréchet (Fr) and slash Fréchet (SFr) distribution for the fixed alpha (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>) and different values of <span class="html-italic">q</span>.</p> Full article ">Figure 3
<p>Graphs of the survival function and hazard function for the SFr distribution with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> and different values of <span class="html-italic">q</span>.</p> Full article ">Figure 4
<p>Skewness coefficient plot of the SFr model (<b>left side</b>). Comparison of the skewness coefficient between SFr and Fr for different values of <span class="html-italic">q</span> (<b>right side</b>).</p> Full article ">Figure 5
<p>Plot of the kurtosis coefficient for the SFr model (<b>left side</b>). Comparison of the kurtosis coefficient between the SFr and Fr models for different values of <span class="html-italic">q</span> (<b>right side</b>).</p> Full article ">Figure 6
<p>Profile of the log-likelihood of the SFr distribution.</p> Full article ">Figure 7
<p>Box plot for the dataset of patients undergoing lung cancer.</p> Full article ">Figure 8
<p>Density adjusted for the dataset of patients undergoing lung cancer in the Fr and SFr distributions.</p> Full article ">Figure 9
<p>QQ plots for the dataset of patients undergoing lung cancer: (<b>a</b>) Fr Model; (<b>b</b>) SFr model.</p> Full article ">Figure 10
<p>Box plot of the dataset of patients undergoing peritoneal dialysis.</p> Full article ">Figure 11
<p>Density adjusted to the dataset of patients undergoing peritoneal dialysis in the Fr, SPN, and SFr distributions.</p> Full article ">Figure 12
<p>QQ plots for the dataset of patients undergoing peritoneal dialysis: (<b>a</b>) Fr model; (<b>b</b>) SPN model; (<b>c</b>) SFr model.</p> Full article ">Figure 13
<p>Profile log-likelihoods of <math display="inline"><semantics> <mi>α</mi> </semantics></math> and <span class="html-italic">q</span> for the dataset of patients undergoing peritoneal dialysis.</p> Full article ">Figure 14
<p>Box plot for the dataset of patients undergoing breast cancer.</p> Full article ">Figure 15
<p>Density adjusted for the dataset of patients undergoing breast cancer in the Fr, SHN, and SFr distributions.</p> Full article ">Figure 16
<p>QQ plots for the dataset of patients undergoing breast cancer: (<b>a</b>) Fr model; (<b>b</b>) SHN model (<b>c</b>) SFr model.</p> Full article ">
19 pages, 14716 KiB  
Article
An Improved Two-Stage Spherical Harmonic ESPRIT-Type Algorithm
by Haocheng Zhou, Zhenghong Liu, Liyan Luo, Mei Wang and Xiyu Song
Symmetry 2023, 15(8), 1607; https://doi.org/10.3390/sym15081607 - 19 Aug 2023
Cited by 2 | Viewed by 1501
Abstract
Sensor arrays are gradually becoming a current research hotspot due to their flexible beam control, high signal gain, robustness against extreme interference, and high spatial resolution. Among them, spherical microphone arrays with complex rotational symmetry can capture more sound field information than planar [...] Read more.
Sensor arrays are gradually becoming a current research hotspot due to their flexible beam control, high signal gain, robustness against extreme interference, and high spatial resolution. Among them, spherical microphone arrays with complex rotational symmetry can capture more sound field information than planar arrays and can convert the collected multiple speech signals into the spherical harmonic domain for processing through spherical modal decomposition. The subspace class direction of arrival (DOA) estimation algorithm is sensitive to noise and reverberation, and its performance can be improved by introducing relative sound pressure and frequency-smoothing techniques. The introduction of the relative sound pressure can increase the difference between the eigenvalues corresponding to the signal subspace and the noise subspace, which is helpful to estimate the number of active sound sources. The eigenbeam estimation of signal parameters via the rotational invariance technique (EB-ESPRIT) is a well-known subspace-based algorithm for a spherical microphone array. The EB-ESPRIT cannot estimate the DOA when the elevation angle approaches 90°. Huang et al. proposed a two-step ESPRIT (TS-ESPRIT) algorithm to solve this problem. The TS-ESPRIT algorithm estimates the elevation and azimuth angles of the signal independently, so there is a problem with DOA parameter pairing. In this paper, the DOA parameter pairing problem of the TS-ESPRIT algorithm is solved by introducing generalized eigenvalue decomposition without increasing the computation of the algorithm. At the same time, the estimation of the elevation angle is given by the arctan function, which increases the estimation accuracy of the elevation angle of the algorithm. The robustness of the algorithm in a noisy environment is also enhanced by introducing the relative sound pressure into the algorithm. Finally, the simulation and field-testing results show that the proposed method not only solves the problem of DOA parameter pairing, but also outperforms the traditional methods in DOA estimation accuracy. Full article
(This article belongs to the Special Issue Symmetry/Asymmetry in Speech and Audio Processing: Topics, Challenges and Advances)
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<p>MAEEs for various source elevation angles and their SNRs. (<b>a</b>) Elevation errors at SNR = 0 dB; (<b>b</b>) Elevation errors at SNR = 20 dB; (<b>c</b>) Azimuth errors at SNR = 0 dB; (<b>d</b>) Azimuth errors at SNR = 20 dB.</p> Full article ">Figure 2
<p>MAEEs of two incoherent sound sources at different SNRs. (<b>a</b>) Azimuth errors at different SNRs; (<b>b</b>) Elevation errors at different SNRs.</p> Full article ">Figure 3
<p>MAEEs of two incoherent sound sources at different snapshots. (<b>a</b>) Azimuth errors at different snapshots; (<b>b</b>) Elevation errors at different snapshots.</p> Full article ">Figure 4
<p>Hash map of NPTS−ESPRIT (each red circle denotes a theoretical DOA, and each blue × denotes an estimated DOA of NPTS−ESPRIT).</p> Full article ">Figure 5
<p>Mean angular error under two different reverberation times.</p> Full article ">Figure 6
<p>Normalized eigenvalues obtained via a singular value decomposition of the source signal covariance matrix (room reverberations T60 = 0.3 s): (<b>a</b>) SNR = 0 dB; (<b>b</b>) SNR = 5 dB; (<b>c</b>) SNR = 10 dB; (<b>d</b>) SNR =15 dB; (<b>e</b>) SNR = 20 dB.</p> Full article ">Figure 7
<p>The overall MAEE about angles under different SNRs with two sources at (70°, 130°) and (170°, 55°).</p> Full article ">Figure 8
<p>MAEE for the angles of interest at (90°, 165°), (−70°, 135°), (130°, 110°), (−30°, 80°), and (170°, 55°) for the presence of five sources with different SNRs.</p> Full article ">Figure 9
<p>Field testing. (<b>a</b>) Spherical microphone array; (<b>b</b>) Conference room.</p> Full article ">
13 pages, 279 KiB  
Article
Impact of Semi-Symmetric Metric Connection on Homology of Warped Product Pointwise Semi-Slant Submanifolds of an Odd-Dimensional Sphere
by Ibrahim Al-Dayel and Meraj Ali Khan
Symmetry 2023, 15(8), 1606; https://doi.org/10.3390/sym15081606 - 19 Aug 2023
Cited by 1 | Viewed by 912
Abstract
Our paper explores warped product pointwise semi-slant submanifolds with a semi-symmetric metric connection in an odd-dimensional sphere and uncovers fundamental results. We also demonstrate how our findings can be applied to the homology of these submanifolds. Notably, we prove that under a specific [...] Read more.
Our paper explores warped product pointwise semi-slant submanifolds with a semi-symmetric metric connection in an odd-dimensional sphere and uncovers fundamental results. We also demonstrate how our findings can be applied to the homology of these submanifolds. Notably, we prove that under a specific condition, there are no stable currents for these submanifolds. This work adds valuable insights into the stability and behavior of warped product pointwise semi-slant submanifolds and sets the foundation for further research in this field. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 2nd Edition)
23 pages, 709 KiB  
Article
Symmetry Analyses of Epidemiological Model for Monkeypox Virus with Atangana–Baleanu Fractional Derivative
by Tharmalingam Gunasekar, Shanmugam Manikandan, Vediyappan Govindan, Piriadarshani D, Junaid Ahmad, Walid Emam and Isra Al-Shbeil
Symmetry 2023, 15(8), 1605; https://doi.org/10.3390/sym15081605 - 19 Aug 2023
Cited by 5 | Viewed by 1825
Abstract
The monkeypox virus causes a respiratory illness called monkeypox, which belongs to the Poxviridae virus family and the Orthopoxvirus genus. Although initially endemic in Africa, it has recently become a global threat with cases worldwide. Using the Antangana–Baleanu fractional order approach, this study [...] Read more.
The monkeypox virus causes a respiratory illness called monkeypox, which belongs to the Poxviridae virus family and the Orthopoxvirus genus. Although initially endemic in Africa, it has recently become a global threat with cases worldwide. Using the Antangana–Baleanu fractional order approach, this study aims to propose a new monkeypox transmission model that represents the interaction between the infected human and rodent populations. An iterative method and the fixed-point theorem are used to prove the existence and uniqueness of the symmetry model’s system of solutions. It shows that the symmetry model has equilibrium points when there are epidemics and no diseases. As well as the local asymptotic stability of the disease-free equilibrium point, conditions for the endemic equilibrium point’s existence have also been demonstrated. For this purpose, the existence of optimal control is first ensured. The aim of the proposed optimal control problem is to minimize both the treatment and prevention costs, and the number of infected individuals. Optimal conditions are acquired Pontryagin’s maximum principle is used. Then, the stability of the symmetry model is discussed at monkeypox-free and endemic equilibrium points with treatment strategies to control the spread of the disease. Numerical simulations clearly show how necessary and successful the proposed combined control strategy is in preventing the disease from becoming epidemic. Full article
(This article belongs to the Special Issue Symmetry Applied in Fractional Dynamics, Fractional Calculus and Inequalities)
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<p>Flowchart of the monkeypox transmission between humans and rodents.</p> Full article ">Figure 2
<p>Susceptible Human.</p> Full article ">Figure 3
<p>Infected Human.</p> Full article ">Figure 4
<p>Treatment Human.</p> Full article ">Figure 5
<p>Recovery Human.</p> Full article ">Figure 6
<p>Susceptible of ordent.</p> Full article ">Figure 7
<p>Infected of ordent.</p> Full article ">Figure 8
<p>Recovery of ordent.</p> Full article ">Figure 9
<p>Susceptible Human.</p> Full article ">Figure 10
<p>Infected Human.</p> Full article ">Figure 11
<p>Treatment Human.</p> Full article ">Figure 12
<p>Recovery Human.</p> Full article ">Figure 13
<p><math display="inline"><semantics> <msub> <mi>u</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>u</mi> <mn>2</mn> </msub> </semantics></math> Recovery.</p> Full article ">Figure 14
<p><math display="inline"><semantics> <msub> <mi>u</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>u</mi> <mn>2</mn> </msub> </semantics></math> <math display="inline"><semantics> <mi>ρ</mi> </semantics></math> (80) Recovery Human.</p> Full article ">Figure 15
<p><math display="inline"><semantics> <msub> <mi>u</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>u</mi> <mn>2</mn> </msub> </semantics></math> <math display="inline"><semantics> <mi>ρ</mi> </semantics></math>(90) Recovery Human.</p> Full article ">
7 pages, 245 KiB  
Article
Hankel Determinant for a Subclass of Starlike Functions with Respect to Symmetric Points Subordinate to the Exponential Function
by Zongtao Li, Dong Guo and Jinrong Liang
Symmetry 2023, 15(8), 1604; https://doi.org/10.3390/sym15081604 - 19 Aug 2023
Cited by 1 | Viewed by 1086
Abstract
Let Ss*(ez) denote the class of starlike functions with respect to symmetric points subordinate to the exponential function, i.e., the functions which satisfy in the unit disk U the condition [...] Read more.
Let Ss*(ez) denote the class of starlike functions with respect to symmetric points subordinate to the exponential function, i.e., the functions which satisfy in the unit disk U the condition 2zf′(z)f(z)−f(−z)≺ez(z∈U). We obtained the sharp estimate of the second-order Hankel determinants H2,3(f) and improved the estimate of the third-order H3,1(f) for this functions class Ss*(ez). Full article
11 pages, 320 KiB  
Review
Antihydrogen and Hydrogen: Search for the Difference
by Ksenia Khabarova, Artem Golovizin and Nikolay Kolachevsky
Symmetry 2023, 15(8), 1603; https://doi.org/10.3390/sym15081603 - 18 Aug 2023
Viewed by 1544
Abstract
Our universe consists mainly of regular matter, while the amount of antimatter seems to be negligible. The origin of this difference, known as the baryon asymmetry, remains undiscovered. Since the discovery of antimatter, many experiments have been carried out to study antiparticles and [...] Read more.
Our universe consists mainly of regular matter, while the amount of antimatter seems to be negligible. The origin of this difference, known as the baryon asymmetry, remains undiscovered. Since the discovery of antimatter, many experiments have been carried out to study antiparticles and to compare matter and antimatter twins. Two of the most sensitive methods in physics, radiofrequency and optical spectroscopy, can be efficiently used to search for the difference. The successful synthesis and trapping of cold antihydrogen atoms opened the possibility of significantly increasing the sensitivity of matter/antimatter tests. This brief review focuses on a hydrogen/antihydrogen comparison using other independent spectroscopic measurements of single particles in traps and other simple atomic systems like positronium. Although no significant difference is detected in today’s level of accuracy, one can push forward the sensitivity by improving the accuracy of 1S–2S positronium spectroscopy, spectroscopy of hyperfine transition in antihydrogen, and gravitational measurements. Full article
(This article belongs to the Section Physics)
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<p>1S and 2S levels energy in H (<math display="inline"><semantics> <mover accent="true"> <mi mathvariant="normal">H</mi> <mo stretchy="false">¯</mo> </mover> </semantics></math>) in external magnetic field. Atoms in states with indexes “c” and “d” can be confined in a magnetic trap. Commonly used transitions for optical two-photon 1S–2S and microwave hyperfine structure spectroscopy are shown.</p> Full article ">
20 pages, 458 KiB  
Article
Fejér-Type Inequalities for Harmonically Convex Functions and Related Results
by Muhammad Amer Latif
Symmetry 2023, 15(8), 1602; https://doi.org/10.3390/sym15081602 - 18 Aug 2023
Cited by 1 | Viewed by 1074
Abstract
In this paper, new Fejér-type inequalities for harmonically convex functions are obtained. Some mappings related to the Fejér-type inequalities for harmonically convex are defined. Properties of these mappings are discussed and, as a consequence, we obtain refinements of some known results. Full article
(This article belongs to the Special Issue Symmetry in Fractional Calculus: Advances and Applications)
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<p>The graph of inequality (<a href="#FD9-symmetry-15-01602" class="html-disp-formula">9</a>) for <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>≤</mo> <mi>κ</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>The graph of inequality (<a href="#FD12-symmetry-15-01602" class="html-disp-formula">12</a>) for <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>≤</mo> <mi>κ</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>The graph of inequality proved in Theorem 20 for <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>≤</mo> <mi>κ</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>The graph of inequality (<a href="#FD19-symmetry-15-01602" class="html-disp-formula">19</a>) for <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>≤</mo> <mi>κ</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>The graph of inequality proved in Theorem 20 for <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>≤</mo> <mi>κ</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>The graph of inequality proved in Theorem 23 for <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>≤</mo> <mi>κ</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">
21 pages, 8587 KiB  
Article
Analysis of Heat Transfer Behavior of Porous Wavy Fin with Radiation and Convection by Using a Machine Learning Technique
by Chandan Kumar, P. Nimmy, Kallur Venkat Nagaraja, R. S. Varun Kumar, Amit Verma, Shalan Alkarni and Nehad Ali Shah
Symmetry 2023, 15(8), 1601; https://doi.org/10.3390/sym15081601 - 18 Aug 2023
Cited by 30 | Viewed by 2172
Abstract
The impact of convection and radiation on the thermal distribution of the wavy porous fin is examined in the present study. A hybrid model that combines the differential evolution (DE) algorithm with an artificial neural network (ANN) is proposed for predicting the heat [...] Read more.
The impact of convection and radiation on the thermal distribution of the wavy porous fin is examined in the present study. A hybrid model that combines the differential evolution (DE) algorithm with an artificial neural network (ANN) is proposed for predicting the heat transfer of the wavy porous fin. The equation representing the thermal variation in the wavy porous fin is reduced to its dimensionless arrangement and is numerically solved using Rung, e-Kutta-Fehlberg’s fourth-fifth order method (RKF-45). The study demonstrates the effectiveness of this hybrid model, and the results indicate that the proposed approach outperforms the ANN model with parameters obtained through grid search (GS), showcasing the superiority of the hybrid DE-ANN model in terms of accuracy and performance. This research highlights the potential of utilizing DE with ANN for improved predictive modeling in the heat transfer sector. The originality of this study is that it addresses the heat transfer problem by optimizing the selection of parameters for the ANN model using the DE algorithm. Full article
(This article belongs to the Special Issue Symmetrical Mathematical Computation in Fluid Dynamics)
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<p>Graphical illustration of a porous wavy fin.</p> Full article ">Figure 2
<p>Workflow of the DE-ANN model.</p> Full article ">Figure 3
<p>Consequences of <math display="inline"><semantics> <mrow> <mi>N</mi> <mi>c</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Θ</mi> <mrow> <mi>W</mi> <mi>F</mi> </mrow> </msub> </mrow> </semantics></math> of solid and porous wavy fin.</p> Full article ">Figure 4
<p>Consequences of <math display="inline"><semantics> <mrow> <mi>N</mi> <mi>r</mi> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Θ</mi> <mrow> <mi>W</mi> <mi>F</mi> </mrow> </msub> </mrow> </semantics></math> of solid and porous wavy fin.</p> Full article ">Figure 5
<p>Consequences of <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> </mrow> </semantics></math> on <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Θ</mi> <mrow> <mi>W</mi> <mi>F</mi> </mrow> </msub> </mrow> </semantics></math> of solid and porous wavy fin.</p> Full article ">Figure 6
<p>Variation in thermal distribution of the porous wavy fin.</p> Full article ">Figure 7
<p>ANN architecture.</p> Full article ">Figure 8
<p>(<b>a</b>) Training prediction plots for GS-ANN (<b>b</b>) Testing prediction plots for GS-ANN.</p> Full article ">Figure 9
<p>(<b>a</b>) Training prediction plots for DE-ANN (<b>b</b>) Testing prediction plots for DE-ANN.</p> Full article ">Figure 10
<p>Loss function evaluation plot of DE-ANN.</p> Full article ">Figure 11
<p>Comparative analysis of GS-ANN and DE-ANN versus <math display="inline"><semantics> <mrow> <mi>N</mi> <mi>c</mi> </mrow> </semantics></math> values.</p> Full article ">Figure 12
<p>Comparative analysis of GS-ANN and DE-ANN versus <math display="inline"><semantics> <mrow> <mi>N</mi> <mi>r</mi> </mrow> </semantics></math> values.</p> Full article ">Figure 13
<p>Comparative analysis of GS-ANN and DE-ANN versus <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>h</mi> </mrow> </semantics></math> values.</p> Full article ">Figure 14
<p>Comparative analysis of GS-ANN and DE-ANN versus <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>t</mi> </msub> </mrow> </semantics></math> values.</p> Full article ">
18 pages, 3206 KiB  
Article
Integrated Optimization of Blocking Flowshop Scheduling and Preventive Maintenance Using a Q-Learning-Based Aquila Optimizer
by Zhenpeng Ge and Hongfeng Wang
Symmetry 2023, 15(8), 1600; https://doi.org/10.3390/sym15081600 - 18 Aug 2023
Viewed by 1491
Abstract
In recent years, integration of production scheduling and machine maintenance has gained increasing attention in order to improve the stability and efficiency of flowshop manufacturing systems. This paper proposes a Q-learning-based aquila optimizer (QL-AO) for solving the integrated optimization problem of blocking flowshop [...] Read more.
In recent years, integration of production scheduling and machine maintenance has gained increasing attention in order to improve the stability and efficiency of flowshop manufacturing systems. This paper proposes a Q-learning-based aquila optimizer (QL-AO) for solving the integrated optimization problem of blocking flowshop scheduling and preventive maintenance since blocking in the jobs processing requires to be considered in the practice manufacturing environments. In the proposed algorithmic framework, a Q-learning algorithm is designed to adaptively adjust the selection probabilities of four key population update strategies in the classic aquila optimizer. In addition, five local search methods are employed to refine the quality of the individuals according to their fitness level. A series of numerical experiments are carried out according to two groups of flowshop scheduling benchmark. Experimental results show that QL-AO significantly outperforms six peer algorithms and two state-of-the-art hybrid algorithms based on Q-Learning on the investigated integrated scheduling problem. Additionally, the proposed Q-learning and local search strategies are effective in improving its performance. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Its Applications to Machine Learning)
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<p>Diagram of blocking flowshop integrated scheduling.</p> Full article ">Figure 2
<p>The flow chart of QL-AO.</p> Full article ">Figure 3
<p>Main effect plots of key parameters. (<b>a</b>) Small-scale instance with 20 jobs and 20 machines; (<b>b</b>) large-scale instance with 400 jobs and 20 machines; (<b>c</b>) large-scale instance with 400 jobs and 20 machines.</p> Full article ">Figure 4
<p>Boxplots of components comparison.</p> Full article ">Figure 5
<p>Probabilities adjusting process.</p> Full article ">Figure 6
<p>Boxplots of optimal solutions.</p> Full article ">Figure 7
<p>Convergence curves of algorithms.</p> Full article ">
34 pages, 428 KiB  
Article
The Quantization of Gravity: The Quantization of the Full Einstein Equations
by Claus Gerhardt
Symmetry 2023, 15(8), 1599; https://doi.org/10.3390/sym15081599 - 17 Aug 2023
Viewed by 1224
Abstract
We quantized the full Einstein equations in a globally hyperbolic spacetime N=Nn+1, n≥3, and found solutions of the resulting hyperbolic equation in a fiber bundle E which can be expressed as a product of [...] Read more.
We quantized the full Einstein equations in a globally hyperbolic spacetime N=Nn+1, n≥3, and found solutions of the resulting hyperbolic equation in a fiber bundle E which can be expressed as a product of spatial eigenfunctions (eigendistributions) and temporal eigenfunctions. The spatial eigenfunctions form a basis in an appropriate Hilbert space while the temporal eigenfunctions are solutions to a second-order ordinary differential equation in R+. In case n≥17 and provided the cosmological constant Λ is negative, the temporal eigenfunctions are eigenfunctions of a self-adjoint operator H^0 such that the eigenvalues are countable and the eigenfunctions form an orthonormal basis of a Hilbert space. Full article
(This article belongs to the Section Physics)
14 pages, 4992 KiB  
Article
A Balanced Symmetrical Branch-Line Microstrip Coupler for 5G Applications
by Salah I. Yahya, Farid Zubir, Leila Nouri, Fawwaz Hazzazi, Zubaida Yusoff, Muhammad Akmal Chaudhary, Maher Assaad, Abbas Rezaei and Binh Nguyen Le
Symmetry 2023, 15(8), 1598; https://doi.org/10.3390/sym15081598 - 17 Aug 2023
Cited by 2 | Viewed by 1414
Abstract
Symmetry in designing a microstrip coupler is crucial because it ensures balanced power division and minimizes unwanted coupling between the coupled lines. In this paper, a filtering branch-line coupler (BLC) with a simple symmetrical microstrip structure was designed, analyzed and fabricated. Based on [...] Read more.
Symmetry in designing a microstrip coupler is crucial because it ensures balanced power division and minimizes unwanted coupling between the coupled lines. In this paper, a filtering branch-line coupler (BLC) with a simple symmetrical microstrip structure was designed, analyzed and fabricated. Based on a mathematical design procedure, the operating frequency was set at 5.2 GHz for WLAN and 5G applications. Moreover, an optimization method was used to improve the performance of the proposed design. It occupied an area of 83.2 mm2. Its harmonics were suppressed up to 15.5 GHz with a maximum level of −15 dB. Meanwhile, the isolation was better than −28 dB. Another advantage of this design was its high phase balance, where the phase difference between its output ports was 270° ± 0.1°. To verify the design method and simulation results, the proposed coupler was fabricated and measured. The results show that all the simulation, design methods, and experimental results are in good agreement. Therefore, the proposed design can be easily used in designing high-performance microstrip-based communication systems. Full article
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<p>Layout and LC circuit of the proposed resonator.</p> Full article ">Figure 2
<p>Layout of the proposed LPFs with their frequency responses.</p> Full article ">Figure 3
<p>The proposed coupler.</p> Full article ">Figure 4
<p>Current density distribution of the proposed coupler at 5.2 GHz.</p> Full article ">Figure 5
<p>S<sub>21</sub> and S<sub>31</sub> as functions of the significant lengths and widths.</p> Full article ">Figure 6
<p><span class="html-italic">S</span><sub>11</sub> and <span class="html-italic">S</span><sub>41</sub> as functions of the significant lengths and widths.</p> Full article ">Figure 7
<p>The steps of our coupler design.</p> Full article ">Figure 8
<p>Simulated and measured frequency responses i.e. S<sub>11,</sub> S<sub>21</sub> (blue lines) and S<sub>41</sub>, S<sub>31</sub> (red lines).</p> Full article ">Figure 8 Cont.
<p>Simulated and measured frequency responses i.e. S<sub>11,</sub> S<sub>21</sub> (blue lines) and S<sub>41</sub>, S<sub>31</sub> (red lines).</p> Full article ">Figure 9
<p>Simulated and measured phase difference between S<sub>21</sub> and S<sub>31</sub>.</p> Full article ">Figure 10
<p>Narrowband frequency responses of the proposed coupler: (<b>a</b>) S<sub>21</sub> and S<sub>31</sub>; (<b>b</b>) S<sub>11</sub> and S<sub>41</sub>.</p> Full article ">Figure 11
<p>Fabricated Coupler.</p> Full article ">
37 pages, 1787 KiB  
Review
Thermal Behavior Modeling of Lithium-Ion Batteries: A Comprehensive Review
by Seyed Saeed Madani, Carlos Ziebert and Mousa Marzband
Symmetry 2023, 15(8), 1597; https://doi.org/10.3390/sym15081597 - 17 Aug 2023
Cited by 10 | Viewed by 7138
Abstract
To enhance our understanding of the thermal characteristics of lithium-ion batteries and gain valuable insights into the thermal impacts of battery thermal management systems (BTMSs), it is crucial to develop precise thermal models for lithium-ion batteries that enable numerical simulations. The primary objective [...] Read more.
To enhance our understanding of the thermal characteristics of lithium-ion batteries and gain valuable insights into the thermal impacts of battery thermal management systems (BTMSs), it is crucial to develop precise thermal models for lithium-ion batteries that enable numerical simulations. The primary objective of creating a battery thermal model is to define equations related to heat generation, energy conservation, and boundary conditions. However, a standalone thermal model often lacks the necessary accuracy to effectively anticipate thermal behavior. Consequently, the thermal model is commonly integrated with an electrochemical model or an equivalent circuit model. This article provides a comprehensive review of the thermal behavior and modeling of lithium-ion batteries. It highlights the critical role of temperature in affecting battery performance, safety, and lifespan. The study explores the challenges posed by temperature variations, both too low and too high, and their impact on the battery’s electrical and thermal balance. Various thermal analysis approaches, including experimental measurements and simulation-based modeling, are described to comprehend the thermal characteristics of lithium-ion batteries under different operating conditions. The accurate modeling of batteries involves explaining the electrochemical model and the thermal model as well as methods for coupling electrochemical, electrical, and thermal aspects, along with an equivalent circuit model. Additionally, this review comprehensively outlines the advancements made in understanding the thermal behavior of lithium-ion batteries. In summary, there is a strong desire for a battery model that is efficient, highly accurate, and accompanied by an effective thermal management system. Furthermore, it is crucial to prioritize the enhancement of current thermal models to improve the overall performance and safety of lithium-ion batteries. Full article
(This article belongs to the Special Issue Symmetry in Electrochemical Process and Application)
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<p>Lithium-ion battery heat-generation (HG) model [<a href="#B21-symmetry-15-01597" class="html-bibr">21</a>].</p> Full article ">Figure 2
<p>(<b>a</b>) Difference between charge and discharge heat losses [<a href="#B21-symmetry-15-01597" class="html-bibr">21</a>], (<b>b</b>–<b>e</b>) the battery surface temperature at different working temperatures and position [<a href="#B22-symmetry-15-01597" class="html-bibr">22</a>], (<b>f</b>) maximum of heat flux at different current rates and temperatures [<a href="#B39-symmetry-15-01597" class="html-bibr">39</a>], and (<b>g</b>) temperature profile inside the battery pack during air cooling [<a href="#B40-symmetry-15-01597" class="html-bibr">40</a>], A: input, B = output.</p> Full article ">Figure 3
<p>(<b>a</b>) Scheme for coupled thermal, electrochemical, and electrical processes interacting in a lithium-ion battery [<a href="#B47-symmetry-15-01597" class="html-bibr">47</a>]; (<b>b</b>) parameter identification methods and models of lithium-ion batteries [<a href="#B48-symmetry-15-01597" class="html-bibr">48</a>].</p> Full article ">Figure 4
<p>(<b>a</b>) Empirical electrothermal modeling of lithium-ion batteries [<a href="#B49-symmetry-15-01597" class="html-bibr">49</a>]; (<b>b</b>) thermal-electrochemical modeling method [<a href="#B44-symmetry-15-01597" class="html-bibr">44</a>].</p> Full article ">
13 pages, 308 KiB  
Article
Three Alternative Model-Building Strategies Using Quasi-Hermitian Time-Dependent Observables
by Miloslav Znojil
Symmetry 2023, 15(8), 1596; https://doi.org/10.3390/sym15081596 - 17 Aug 2023
Cited by 1 | Viewed by 999
Abstract
In the conventional (so-called Schrödinger-picture) formulation of quantum theory the operators of observables are chosen self-adjoint and time-independent. In the recent innovation of the theory, the operators can be not only non-Hermitian but also time-dependent. The formalism (called non-Hermitian interaction-picture, NIP) requires a [...] Read more.
In the conventional (so-called Schrödinger-picture) formulation of quantum theory the operators of observables are chosen self-adjoint and time-independent. In the recent innovation of the theory, the operators can be not only non-Hermitian but also time-dependent. The formalism (called non-Hermitian interaction-picture, NIP) requires a separate description of the evolution of the time-dependent states ψ(t) (using Schrödinger-type equations) as well as of the time-dependent observables Λj(t), j=1,2,…,K (using Heisenberg-type equations). In the unitary-evolution dynamical regime of our interest, both of the respective generators of the evolution (viz., in our notation, the Schrödingerian generator G(t) and the Heisenbergian generator Σ(t)) have, in general, complex spectra. Only the spectrum of their superposition remains real. Thus, only the observable superposition H(t)=G(t)+Σ(t) (representing the instantaneous energies) should be called Hamiltonian. In applications, nevertheless, the mathematically consistent models can be based not only on the initial knowledge of the energy operator H(t) (forming a “dynamical” model-building strategy) but also, alternatively, on the knowledge of the Coriolis force Σ(t) (forming a “kinematical” model-building strategy), or on the initial knowledge of the Schrödingerian generator G(t) (forming, for some reason, one of the most popular strategies in the literature). In our present paper, every such choice (marked as “one”, “two” or “three”, respectively) is shown to lead to a construction recipe with a specific range of applicability. Full article
(This article belongs to the Special Issue Physics and Symmetries of Commutative and Noncommutative Quantum Field Theory)
12 pages, 2489 KiB  
Article
Symmetry Function in Trans-Tibial Amputees Gait Supplied with the New Concept of Affordable Dynamic Foot Prosthesis—Case Study
by Michal Murawa, Jakub Otworowski, Sebastian But, Jaroslaw Kabacinski, Lukasz Kubaszewski and Adam Gramala
Symmetry 2023, 15(8), 1595; https://doi.org/10.3390/sym15081595 - 17 Aug 2023
Cited by 2 | Viewed by 1377
Abstract
The development of modern technologies has made it much easier to regain the ability to walk after losing a lower limb. The variety of prosthetic feet available on the market allows for optimal choice and appropriate adjustment of the foot prosthesis to the [...] Read more.
The development of modern technologies has made it much easier to regain the ability to walk after losing a lower limb. The variety of prosthetic feet available on the market allows for optimal choice and appropriate adjustment of the foot prosthesis to the trans-tibial amputee patient’s needs. Unfortunately, the best solutions are often not available to everyone due to their high prices. This study compares the gait patterns of patients using the new concept of an affordable dynamic foot with those of other commonly available but much more expensive foot prostheses. The kinematic and spatio-temporal parameters of gait obtained using the motion capture system were analyzed. For a clear picture of changes in bilateral deficits during gait for the pelvis, hip, knee, and ankle joints, the symmetry function was used. The results indicate that the new and cheaper concept of foot prostheses offers a very similar level of gait quality to that provided by more expensive and popular solutions. The authors suggest that the use of symmetry function thresholds of 10% does not work for amputees. Full article
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<p>SF between the ILE side and the ULE side for ankle, knee, hip, and pelvic motion in sagittal plane in the gait cycle. The black and blue lines represent FP and DFP, respectively. Areas of +/−0% to 5%, +/−5% to 10%, and +/−10 to 15% SF are marked in grading shades of red (from the lightest to the darkest).</p> Full article ">Figure 2
<p>SF between the ILE side and the ULE side for hip and pelvic motion in the frontal and transverse plane in the gait cycle. The black and blue lines represent FP and DFP, respectively. Areas of +/−0% to 5%, +/−5% to 10%, and +/−10 to 15% SF are marked in grading shades of red (from the lightest to the darkest).</p> Full article ">
24 pages, 1254 KiB  
Article
An Evaluation Model for the Influence of KOLs in Short Video Advertising Based on Uncertainty Theory
by Meiling Jin, Yufu Ning, Fengming Liu, Fangyi Zhao, Yichang Gao and Dongmei Li
Symmetry 2023, 15(8), 1594; https://doi.org/10.3390/sym15081594 - 17 Aug 2023
Cited by 5 | Viewed by 3759
Abstract
In the era of rapid growth in the short video industry, it is very important to find more accurate suitable advertising promoters, namely Key Opinion Leaders, to promote the development of short video commerce. A mathematical method is needed to grade and evaluate [...] Read more.
In the era of rapid growth in the short video industry, it is very important to find more accurate suitable advertising promoters, namely Key Opinion Leaders, to promote the development of short video commerce. A mathematical method is needed to grade and evaluate KOL’s abilities. Only in this way can advertisers better determine the value of KOL and determine whether it is suitable for promoting its products. Moreover, in the hierarchical evaluation of KOL, there is not only structured and quantifiable information, but also a large amount of unstructured and linguistic non-quantifiable information. Therefore, this article regards unquantifiable information as an uncertain variable and uses a comprehensive evaluation method based on uncertainty theory to handle subjective uncertainty in the evaluation process. Among them, all uncertain variables are symmetric. The main contribution of this article is the provision of a new evaluation method for KOL grading. Firstly, a two-level evaluation index system for KOL was established. Secondly, the importance and annotation of the Index set are set as uncertain variables, and the KOL evaluation model is constructed. Finally, two KOLs on TikTok were selected for comparative analysis to determine the importance ranking and KOL scores of each level of indicator, verifying the effectiveness and practicality of this method. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—Volume II)
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<p>Distribution of evaluative data of primary level indicators for Zhu Xiaohan.</p> Full article ">Figure 2
<p>Distribution of evaluative data for secondary level indicators.</p> Full article ">Figure 3
<p>Distribution of evaluative data of primary level indicators for Fang Qi.</p> Full article ">Figure 4
<p>Distribution of evaluative data for primary level indicators.</p> Full article ">
14 pages, 4870 KiB  
Article
Rippled Graphene as an Ideal Spin Inverter
by Ján Buša, Michal Pudlák and Rashid Nazmitdinov
Symmetry 2023, 15(8), 1593; https://doi.org/10.3390/sym15081593 - 16 Aug 2023
Cited by 2 | Viewed by 1228
Abstract
We analyze a ballistic electron transport through a corrugated (rippled) graphene system with a curvature-induced spin–orbit interaction. The corrugated system is connected from both sides to two flat graphene sheets. The rippled structure unit is modeled by upward and downward curved surfaces. The [...] Read more.
We analyze a ballistic electron transport through a corrugated (rippled) graphene system with a curvature-induced spin–orbit interaction. The corrugated system is connected from both sides to two flat graphene sheets. The rippled structure unit is modeled by upward and downward curved surfaces. The cooperative effect of N units connected together (the superlattice) on the transmission of electrons that incident at the arbitrary angles on the superlattice is considered. The set of optimal angles and corresponding numbers of N units that yield the robust spin inverter phenomenon are found. Full article
(This article belongs to the Special Issue Cooperative Effects in Finite Systems)
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<p>(<b>a</b>) Sketch of the superlattice. The ballistic electron, coming from the left of the flat graphene sheet, incidents on the superlattice structure at an arbitrary angle <math display="inline"><semantics> <mi>φ</mi> </semantics></math>. (<b>b</b>) Cross-section of the system that consists of two flat graphene sheets and superlattice. The two flat surfaces are the region L, defined in the intervals <math display="inline"><semantics> <mrow> <mo>−</mo> <mo>∞</mo> <mo>&lt;</mo> <mi>x</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math>; and the region R, defined in the intervals <math display="inline"><semantics> <mrow> <mn>4</mn> <mi>N</mi> <mi>R</mi> <mo form="prefix">cos</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>&lt;</mo> <mi>x</mi> <mo>&lt;</mo> <mo>∞</mo> </mrow> </semantics></math>. The region I (the concave arc) is a part of a nanotube of radius <span class="html-italic">R</span>, defined as <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>x</mi> <mo>&lt;</mo> <mn>2</mn> <mi>R</mi> <mo form="prefix">cos</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> </mrow> </semantics></math>. At <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, the up surface is half that of the nanotube, while at <math display="inline"><semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> the curvature does not exist. For the sake of analysis, we introduce the angle <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> <mo>−</mo> <mn>2</mn> <msub> <mi>θ</mi> <mn>0</mn> </msub> </mrow> </semantics></math>. The region II (the convex arc with the radius <span class="html-italic">R</span>) is characterized by similar parameters to those of region I. Here, we have <math display="inline"><semantics> <mrow> <mo>−</mo> <mo>∞</mo> <mo>&lt;</mo> <mi>y</mi> <mo>&lt;</mo> <mo>∞</mo> </mrow> </semantics></math>. To describe the scattering phenomenon, one has to define wave functions in different regions: flat (L,R) and curved (I, II) graphene surfaces.</p> Full article ">Figure 2
<p>The energy spectrum (<a href="#FD14-symmetry-15-01593" class="html-disp-formula">14</a>) as a function of the magnetic quantum number m. For a given energy <span class="html-italic">E</span>, the magnetic quantum number values <math display="inline"><semantics> <msub> <mi>m</mi> <mo>−</mo> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>m</mi> <mo>+</mo> </msub> </semantics></math> are determined from the crossing points of the dashed and solid lines by the horizontal line (<span class="html-italic">E</span>), presented as an example. The results are obtained at <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math> Å, <math display="inline"><semantics> <mrow> <mi>ϕ</mi> <mo>=</mo> <mi>π</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </semantics></math>. The values of spin–orbital strengths <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mi>x</mi> </msub> <mo>≈</mo> <mn>0.267</mn> </mrow> </semantics></math> eV, <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mi>y</mi> </msub> <mo>=</mo> <mn>0.00355</mn> </mrow> </semantics></math> eV (see Equation (<a href="#FD10-symmetry-15-01593" class="html-disp-formula">10</a>)) follow from the values of the parameters <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>(</mo> <mn>4.5</mn> <mo>×</mo> <mn>1.42</mn> <mo>)</mo> </mrow> </semantics></math> eV·Å, <math display="inline"><semantics> <mrow> <msup> <mi>γ</mi> <mo>′</mo> </msup> <mo>=</mo> <mfrac> <mn>8</mn> <mn>3</mn> </mfrac> <mi>γ</mi> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>The maximal spin-flip probabilities <math display="inline"><semantics> <msub> <mi>P</mi> <mrow> <mi>N</mi> <mo>,</mo> <mi>φ</mi> </mrow> </msub> </semantics></math> in the superlattice for various combinations <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>N</mi> <mo>,</mo> <mi>φ</mi> <mo>}</mo> </mrow> </semantics></math> (see discussion below) in the energy interval <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>=</mo> <mn>0.02</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>1</mn> </mrow> </semantics></math> eV for <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math> Å. For the sake of illustration, the points <math display="inline"><semantics> <msubsup> <mi>P</mi> <mrow> <mi>N</mi> <mo>,</mo> <mi>φ</mi> </mrow> <mrow> <mrow> <mo>{</mo> <mi>i</mi> </mrow> <mo>}</mo> </mrow> </msubsup> </semantics></math> for corresponding energies <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mi>i</mi> </msub> <mo>=</mo> <mi mathvariant="sans-serif">Δ</mi> <mi>E</mi> <mo>×</mo> <msub> <mi>N</mi> <mi>i</mi> </msub> </mrow> </semantics></math> (<math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mi>E</mi> <mo>=</mo> <mn>0.02</mn> </mrow> </semantics></math> eV, <math display="inline"><semantics> <mrow> <msub> <mi>N</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>50</mn> </mrow> </semantics></math>) are connected by a solid line.</p> Full article ">Figure 4
<p>The spin inversion probabilities <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mrow> <mi>N</mi> <mo>,</mo> <mi>φ</mi> </mrow> </msub> <mo>≥</mo> <mn>0.99</mn> </mrow> </semantics></math> (yellow domain) as a function of number units <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>50</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>450</mn> </mrow> </semantics></math> in the superlattice for angles <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <msup> <mn>5</mn> <mo>°</mo> </msup> <mo>,</mo> <msup> <mn>5.01</mn> <mo>°</mo> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mn>37</mn> <mo>°</mo> </msup> </mrow> </semantics></math>, for <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math> Å, at the incident energy beam <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> eV (<b>left</b>) and <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> eV (<b>right</b>).</p> Full article ">Figure 5
<p>Spin-flip points on the <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>×</mo> <mi>φ</mi> </mrow> </semantics></math> mesh: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math> Å, <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> eV; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>18</mn> </mrow> </semantics></math> Å, <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics></math> eV; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>24</mn> </mrow> </semantics></math> Å, <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> eV; (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>36</mn> </mrow> </semantics></math> Å, <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> eV.</p> Full article ">
13 pages, 280 KiB  
Article
Conditional Uncertainty Distribution of Two Uncertain Variables and Conditional Inverse Uncertainty Distribution
by Lihui Wang, Yufu Ning, Xiumei Chen, Shukun Chen and Hong Huang
Symmetry 2023, 15(8), 1592; https://doi.org/10.3390/sym15081592 - 16 Aug 2023
Viewed by 1007
Abstract
It is noted that some uncertain variables are independent while others are not. In general, there is a symmetrical relationship between independence and dependence among uncertain variables. The utilization of conditional uncertain measures as well as conditional uncertainty distributions proves highly efficacious in [...] Read more.
It is noted that some uncertain variables are independent while others are not. In general, there is a symmetrical relationship between independence and dependence among uncertain variables. The utilization of conditional uncertain measures as well as conditional uncertainty distributions proves highly efficacious in resolving uncertainties pertaining to an event subsequent to the acquisition of knowledge about other events. In this paper, the theorem about the conditional uncertainty distribution of two uncertain variables is proposed. It is demonstrated that the theorem holds regardless of whether the two variables are independent or not. In addition, it is also found that uncertainty distribution possesses an inherent inverse function when it is a regular uncertainty distribution within the framework of Uncertainty Theory; therefore, this paper delves into investigating the conditional inverse uncertainty distribution, including specific cases of the conditional inverse uncertainty distributions. Meanwhile, illustrative examples are applied to clarify the findings. Full article
(This article belongs to the Special Issue Fuzzy Set Theory and Uncertainty Theory—Volume II)
22 pages, 773 KiB  
Article
Theoretical Evaluation of the Reinjection Probability Density Function in Chaotic Intermittency
by Sergio Elaskar and Ezequiel del Río
Symmetry 2023, 15(8), 1591; https://doi.org/10.3390/sym15081591 - 16 Aug 2023
Viewed by 1147
Abstract
The traditional theory of chaotic intermittency developed for return maps hypothesizes a uniform density of reinjected points from the chaotic zone to the laminar one. In the past few years, we have described how the reinjection probability density function (RPD) can be generalized [...] Read more.
The traditional theory of chaotic intermittency developed for return maps hypothesizes a uniform density of reinjected points from the chaotic zone to the laminar one. In the past few years, we have described how the reinjection probability density function (RPD) can be generalized as a power law function. Here, we introduce a broad and general analytical approach to determine the RPD function and other statistical variables, such as the characteristic relation traditionally utilized to characterize the chaotic intermittency type. The proposed theoretical methodology is simple to implement and includes previous studies as particular cases. It is compared with numerical data, the M function methodology, and the Perron–Frobenius technique, showing high accuracy between them. Full article
(This article belongs to the Section Mathematics)
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Figure 1

Figure 1
<p>Map provided by Equation (<a href="#FD50-symmetry-15-01591" class="html-disp-formula">50</a>) for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>. Blue line: the map. Dashed black line: the bisector.</p> Full article ">Figure 2
<p>Map (<a href="#FD50-symmetry-15-01591" class="html-disp-formula">50</a>). Comparison between the RPD functions for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0.025</mn> </mrow> </semantics></math>. Red points: numerical RPD. Dashed green line: RPD calculated by the <span class="html-italic">M</span> function methodology. Blue line: theoretical RPD evaluated by Equation (<a href="#FD60-symmetry-15-01591" class="html-disp-formula">60</a>).</p> Full article ">Figure 3
<p>Map provided by Equation (<a href="#FD68-symmetry-15-01591" class="html-disp-formula">68</a>), <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.0001</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>. Blue line: the map. Dashed black line: the bisector.</p> Full article ">Figure 4
<p>Equation (<a href="#FD68-symmetry-15-01591" class="html-disp-formula">68</a>). Comparison between the RPD functions for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.0001</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0.02</mn> </mrow> </semantics></math>. Red points: numerical RPD. Dashed green line: RPD calculated by the <span class="html-italic">M</span> function methodology. Black line: theoretical RPD with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>/</mo> <mn>4</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Map provided by Equation (<a href="#FD78-symmetry-15-01591" class="html-disp-formula">78</a>) for <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>0.65</mn> </mrow> </semantics></math>. Blue line: the map. Dashed black line: the bisector.</p> Full article ">Figure 6
<p>Comparison between the RPD functions for <math display="inline"><semantics> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>2</mn> <mo>/</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mi>z</mi> </msub> <mo>=</mo> <mn>0.65</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> </mrow> </semantics></math> 250,000. Red points: numerical RPD. Dashed blue line: RPD calculated by the <span class="html-italic">M</span> function methodology using two contributions. Dashed magenta line: RPD calculated by the <span class="html-italic">M</span> function methodology using one contribution. Green points: theoretical RPD calculated using the Perron–Frobenius operator. Black line: theoretical evaluation using Equations (<a href="#FD96-symmetry-15-01591" class="html-disp-formula">96</a>) and (<a href="#FD97-symmetry-15-01591" class="html-disp-formula">97</a>).</p> Full article ">
10 pages, 1606 KiB  
Article
Evaluation of the Reduced Protocol for the Assessment of Rate of Force Development Scaling Factor
by Života Stefanović, Filip Kukić, Olivera M. Knežević, Nejc Šarabon and Dragan M. Mirkov
Symmetry 2023, 15(8), 1590; https://doi.org/10.3390/sym15081590 - 16 Aug 2023
Cited by 1 | Viewed by 1198
Abstract
The rate of force development scaling factor (RFD-SF) has been used to assess neuromuscular quickness. However, the common protocols are lengthy. This study evaluated the validity and reliability of the reduced protocol to assess the RFD-SF and its validity in detecting inter-limb asymmetries. [...] Read more.
The rate of force development scaling factor (RFD-SF) has been used to assess neuromuscular quickness. However, the common protocols are lengthy. This study evaluated the validity and reliability of the reduced protocol to assess the RFD-SF and its validity in detecting inter-limb asymmetries. Eighteen participants (five females and thirteen males; mean age = 20.8 ± 0.6 years) performed the common and reduced RFD-SF protocols (five isometric pulse knee extensions at 30 and 70% of maximal voluntary contraction). A repeat measure design was employed including one test session of the common protocol and two test sessions of the reduced protocol. Correlation analysis was conducted to investigate the association between the two protocols, while a paired-sample t-test and a Bland–Altman plot assessed whether the reduced protocol provided valid results. The between-day reliability was assessed using an intra-class correlation coefficient, coefficient of variation, typical error of measurement, and paired-sample t-test. The validity to detect asymmetries was checked with the paired-sample t-test. The correlation between RFD-SF obtained using two protocols was significant (p < 0.001) and very large for the dominant (r = 0.71) and non-dominant (r = 0.80) legs. No significant difference occurred between protocols in the RFD-SF for the dominant (p = 0.480, d = 0.17) and non-dominant legs (p = 0.213, d = 0.31). The reliability was acceptable for both legs, with no between-day difference for the dominant (p = 0.393) and non-dominant legs (p = 0.436). No significant difference between the two protocols (p = 0.415, d = 0.19) was found in the detection of inter-limb asymmetries. The results of this study suggest that the reduced protocol could be used as a valid and reliable alternative to the common protocol, as well as to identify interlimb asymmetries. Full article
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Figure 1

Figure 1
<p>Testing setup. Participant in a custom-made chair (1: force transducer; 2: shanks; 3: rigid straps; 4: acquisition and analog-to-digital conversion unit; 5: monitor with visual feedback).</p> Full article ">Figure 2
<p>Distribution of R<sup>2</sup> for Fmax–RFDmax associations obtained in two protocols.</p> Full article ">Figure 3
<p>The distribution of participants and the sample means for the common and reduced RFD-SF protocols. Note: Difference = difference in RFD-SF obtained in common and reduced protocols, as obtained with paired samples <span class="html-italic">t</span>-test.</p> Full article ">Figure 4
<p>The Bland–Altman plot for the agreement between the two protocols.</p> Full article ">Figure 5
<p>The asymmetries obtained using the common and reduced protocol.</p> Full article ">
9 pages, 265 KiB  
Article
New Results about Fuzzy Differential Subordinations Associated with Pascal Distribution
by Sheza M. El-Deeb and Luminiţa-Ioana Cotîrlă
Symmetry 2023, 15(8), 1589; https://doi.org/10.3390/sym15081589 - 15 Aug 2023
Cited by 2 | Viewed by 872
Abstract
Based upon the Pascal distribution series [...] Read more.
Based upon the Pascal distribution series Nq,λr,mΥ(ζ):=ζ+∑j=m+1∞j+r−2r−11+λ(j−1)qj−1(1−q)rajζj, we can obtain a set of fuzzy differential subordinations in this investigation. We also newly obtain class Pq,λF,r,mη of univalent analytic functions defined by the operator Nq,λr,m, give certain properties for the class Pq,λF,r,mη and also obtain some applications connected with a special case for the operator. New research directions can be taken on fuzzy differential subordinations associated with symmetry operators. Full article
(This article belongs to the Special Issue Symmetry in Pure Mathematics and Real and Complex Analysis)
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